birthday paradox

The birthday paradox is the fact – not really a paradox – that you need a group of only 23 people for there to be a better than 50:50 chance that two of these people will have the same birthday. This seems surprising because we are used to comparing our particular birthdays with others and only rarely finding a perfect match. The probability of any two individuals having the same birthday is just 1/365. Even if you were to ask 20 people, the probability of finding someone with your birthday is still less than 1/20. But the odds improve dramatically when a group of people ask each other about their birthdays because then there are many more opportunities for a match-up. One way to calculate the probability of a birthday match is to count the pairs of people involved. In a room of 23 people, there are (23 × 22)/2, or 253, possible pairs. Each pair has a probability of success of 1/365 = 0.00274 (0.274%), and thus a probability of failure of (1 - 0.00274) = 0.99726 (99.726%). The probability of no match among any of the pairs of people is 0.99726 to the 253th power, which is 0.499 (49.9%). So the probability of a successful match is (1 - 0.499), or slightly better than evens. With 42 people, the probability of a birthday match climbs to 90%.